Problem: Solve for $n$. $\left(z^2\right)^{n}=z^{10}$ $n=$
The general rule for powers of powers is $\left(x^m\right)^{n}=x^{m\cdot n}$. Let's expand the powers for $ \left({z^2}\right)^{{n}}=z^{10}}$. $\begin{aligned} \left({z^2}\right)^{5}&=\underbrace{{z^2\cdot z^2 \cdot z^2 \cdot z^2 \cdot z^2}}_{n\text{ times}} \\\\\\ &=\underbrace{ \underbrace{{z\cdot z}}_\text{2 times} \cdot \underbrace{{z\cdot z}}_\text{2 times} \cdot \underbrace{{z\cdot z}}_\text{2 times} \cdot \underbrace{{z\cdot z}}_\text{2 times} \cdot \underbrace{{z\cdot z}}_\text{2 times}} _{n\text{ times}} \\\\ &=\underbrace{z\cdot z\cdot z\cdot z\cdot z\cdot z \cdot z \cdot z \cdot z \cdot z}_\text{10 times}} \\\\ \end{aligned}$ $n = 5$